Round and Hollow Shafts.
419
The first formula may be represented by the lamina at a, and
the second by that at <£, and the total of the stresses on all the
rings will be given by the pyramid. Again :
Moment of stress at ring r = a x r
Moment of stress at ring ^ = b x ^
. *. Moment of all the stresses = contents of pyramid x average arm
== (base x \ height) x (f height)
.*. Moment of resistance of section =/s--------x - r =/s----
------------------------------------ /s 3 4 y 2
(and putting - = r) = /s ( 7- J
Strength of Hollow Round Shaft. Fig. 371 shews a
shaft of diameters D and d, externally and internally respectively.
At radius R" the stress is /s, but at radius r it is proportionately
less, being but fs
=-.
The strength of the hollow shaft will be
as stressed in situ,
TTD3 // A7*"d* TT /D4-aft\
"TJT "~ VSD/ ~i6~ ""' 16 \ D /
found by deducting the strength of shaft
from the strength of a solid shaft D.
Moment of resistance = mom1 of solid shaft minus mom* of core
(See Apj>. ///.,
/ 921.)
Strength of Square Shaft.In this case we shall not
use the previous methods, but shall adopt a construction which,
although requiring careful drawing, can be employed for any
section, and is therefore general. In Fig. 372, A B CD is the sjiaft
section, divided into concentric rings as before. Erect perpen-
diculars on ED to represent the length of every ring, and bound
these by the figure E G L F D. j F = ?r s, and F E is a straight line,
while the lengths between F and D are found by stepping off each
set of four arcs with dividers. Now the stress will be greatest
at D, and will decrease gradually to zero at E, and the product
(f& x ring area) will proportionately decrease, so the total stress
may be obtained by imagining^ to be constant, and each ring
to have a value represented by the circumference decreased